E. Jacquelin Dietz

A Nonparametric Multivariate Test for Monotone Trend with Pharmaceutical Applications

Abstract During a drug trial, each subject may have many blood constituents measured at regular time intervals. The traditional method of evaluating such data using normal ranges has undergone much controversy in the clinical chemistry literature. We propose a new multivariate test for monotone trend based on Kendalls τ statistic as a way to test for change in blood constituents over the course of a drug experiment. The new test avoids the unrealistic assumptions and repeated testing problem of the normal range method. An example illustrates the new procedure.

Teaching Regression in a Nonparametric Statistics Course

Abstract Various estimators of slope, intercept, and mean response in the simple linear regression problem are compared in terms of unbiasedness, efficiency, breakdown, and mean squared error. Theils estimator of slope and two intercept estimators based on Theils estimator are recommended for inclusion in nonparametrics courses as robust, efficient, and easy-to-calculate alternatives to least squares.

A Cooperative Learning Activity on Methods of Selecting a Sample

Abstract This article describes a classroom activity designed to stimulate students to think creatively about methods of selecting a representative sample from a population. Students are presented with a dataset consisting of gender, SAT verbal score, SAT mathematics score, and high school grade point average for 317 freshmen from North Carolina State University. The students, who have not yet studied sampling, work in groups of three or four to generate three possible methods for selecting a representative sample of 20 freshmen from the population of 317. Each group uses its proposed methods to select three samples and computes various summary statistics and plots for the variables in each sample. The students are then given corresponding information for the entire population. After comparing sample statistics and population parameters, the groups evaluate the advantages and disadvantages of the proposed sampling methods. During the two semesters that I have used this activity in my Statistics 101 class,...

A comparison of robust estimators in simple linear regression

The mean squared errors of various estimators of slope, intercept, and mean response in the simple linear regression problem are compared in a simulation study. A weighted median estimator of slope proposed by Sievers (1978) and Scholz (1978) and two intercept estimators based upon it are found to perform well for most error distributions studied. Theils (1950) estimator of slope and two intercept estimators based on it are preferable for certain heavily contaminated error distributions.

Multivariate generalizations of jonckheere's test for ordered alternatives

In many dose-response studies, each of several independent groups of animals is treated with a different dose of a substance. Many response variables are then measured on each animal. The distributions of the response variables may be nonnormal, and Jonckheeres (1954) test for ordered alternatives in the one-way layout is sometimes used to test whether the level of a single variable increases with increasing dose. In some applications, however, it is important to consider a set of response variables simultaneously. For instance, an increase in each of certain enzymes in the blood serum may suggest liver damage. To test whether these enzyme levels increase with increasing dose, it may be preferable to consider these enzymes as a group, rather than individually. I propose two multivariate generalizations of Jonckheeres univariate test. Each multivariate test statistic is a function of coordinate-wise Jonckheere statistics—one a sum, the other a quadratic form. The sum statistic can be used to test the alt...

Bivariate Nonparametric Tests for the One-Sample Location Problem

Abstract Many nonparametric competitors to Hotellings T2 test lack T 2s property of invariance under linear transformations. Such tests have power and efficiency that depend on the direction of shift and the covariance matrix of the alternative distribution. This article modifies a previously proposed bivariate sign test and signed rank test to overcome this difficulty. Asymptotic results and a Monte Carlo study indicate that the efficiency and small-sample power of the new procedures compare favorably to those of the original tests. An example illustrates the application of the new procedures.

The Rank Sum Test in the Linear Logistic Model

Abstract A simple modification of the rank sum test that can be carried out quickly and easily with pencil and paper is shown to be competitive in terms of power with the much more laborious likelihood ratio test for testing the hypothesis of zero slope in the simple linear logistic model.

Agreement probabilities for some CPIT—neyman smooth tests

The CPIT transformations do not, in general, give the same set of transformed values for different orderings of the observations in a sample. When these transformations are followed by a test of uniformity to give an overall goodness—of—fit test, it is possible to obtain different results for different orderings of a sample. We consider here the probability that two goodness—of—fit tests based on randomly selected permutations of the same sample and a Neyman smooth uniformity test will agree in their conclusions. We observe for the cases considered in a simulation study that the probability of agreement is generally large and tends to one with increasing n, under alternative hypotheses.